08/12/2013, 10:17 PM

Well I've been muddling this idea around for a while. I have been trying to create a hyper operator space and I recently realized the form of this. I'll start as follows:

If is a hyper operator then, is a hyper operator created by forming left composition. I.e:

for all then

Associate to every function that is a finite product a number as follows:

Where and p_n is the nth prime.

Now hyper operator space is the following:

Now define the inner product as follows:

Where quite clearly (f,f) converges for all elements since the terms decay to zero across x and y faster or just as fast as

We say all the functions are dense in

Orthonormalize them to get such that:

Now we have the advantage of being in a Hilbert space and having an orthonormal basis.

The first operator we have is the transfer operator:

Since this operator is well defined for any element of where

Suppose:

exists such that for all values that [s] returns natural numbers at, this is our solution to hyper operators.

I think the key is to invesetigate the inner product.

If is a hyper operator then, is a hyper operator created by forming left composition. I.e:

for all then

Associate to every function that is a finite product a number as follows:

Where and p_n is the nth prime.

Now hyper operator space is the following:

Now define the inner product as follows:

Where quite clearly (f,f) converges for all elements since the terms decay to zero across x and y faster or just as fast as

We say all the functions are dense in

Orthonormalize them to get such that:

Now we have the advantage of being in a Hilbert space and having an orthonormal basis.

The first operator we have is the transfer operator:

Since this operator is well defined for any element of where

Suppose:

exists such that for all values that [s] returns natural numbers at, this is our solution to hyper operators.

I think the key is to invesetigate the inner product.